Marigold has a lot of tomato plants in her vegetable garden. Marigold is planning to pick the ripe tomatoes and make salsa with them. She looks up a basic recipe. The recipe says that for every cup of tomatoes, she will need 0.5 onion. For every onion, she needs 4 cloves of garlic. How can Marigold determine how many cloves of garlic she needs in terms of the number of cups of tomatoes she picks?

In this concept, you will learn to identify and use the commutative and associative properties of multiplication with decimals.

Commutative and Associative Properties of Multiplication with Decimals

The Commutative Property of Multiplication states that when finding a product, changing the order of the factors will not change their product. In symbols, the Commutative Property of Multiplication says that for numbers \begin{align*}a\end{align*} and \begin{align*}b\end{align*}:

\begin{align*}a \cdot b=b \cdot a\end{align*}

Here is an example using simple whole numbers.

Show that \begin{align*}2 \cdot 4 = 4 \cdot 2\end{align*}.

First, find \begin{align*}2 \cdot 4\end{align*}.

\begin{align*}2 \cdot 4 = 8\end{align*}

Next, find \begin{align*}4 \cdot 2\end{align*}.

\begin{align*}4 \cdot 2 = 8\end{align*}

Notice that both products are 8.

The answer is that because both \begin{align*}2 \cdot 4\end{align*} and \begin{align*}4 \cdot 2\end{align*} are equal to 8, they are equal to each other.

\begin{align*}2 \cdot 4 = 4 \cdot 2\end{align*}

The Associative Property of Multiplication states that when finding a product, changing the way factors are grouped will not change their product. In symbols, the Associative Property of Multiplication says that for numbers \begin{align*}a\end{align*},\begin{align*}b\end{align*} and \begin{align*}c\end{align*}:

The answer is that because both \begin{align*}(2 \cdot 5) \cdot 6\end{align*} and \begin{align*}2 \cdot (5 \cdot 6)\end{align*} are equal to 60, they are equal to each other.

\begin{align*}(2 \cdot 5) \cdot 6=2 \cdot (5 \cdot 6)\end{align*}

Both the Commutative Property of Multiplication and the Associative Property of Multiplication can be useful in simplifying expressions. The Commutative Property of Multiplication allows you to reorder factors while the Associative Property of Multiplication allows you to regroup factors.

Here is an example.

Simplify \begin{align*}29.3(12.4x)\end{align*}.

First, use the Associative Property of Multiplication to regroup the factors.

The answer is that \begin{align*}(0.3x) \cdot 0.4\end{align*} simplifies to \begin{align*}0.12x\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Marigold, who is planning to make salsa.

Her recipe says that for every cup of tomatoes she will need 0.5 onion, and for every onion she will need 4 cloves of garlic. Marigold wants to figure out how many cloves of garlic she will need in terms of the number of cups of tomatoes she picks.

First, Marigold should write an expression for this situation. She should start by defining her variable. She doesn’t know how many cups of tomatoes she will have, so that unknown quantity will be her variable.

Let \begin{align*}x\end{align*} equal the number of cups of tomatoes Marigold picks.

Now, the problem says that she will need 0.5 onion for every tomato. So the number of onions she needs is \begin{align*}0.5x\end{align*}.

Next, the problem says that she will need 4 cloves of garlic for every onion. Since she will have \begin{align*}0.5x\end{align*} onions, she will need \begin{align*}(0.5x) \cdot 4\end{align*} cloves of garlic.

Now, Marigold can simplify the expression.

First, she can use the Commutative Property of Multiplication to reorder the factors.